Bottom Line:
We applied a stability analysis and identified rules for choosing biologically tunable parameter relationships to increase the likelihood of successful patterning.Greater steepness increases parameter space and even reduces the requirement for differential diffusion between activator and inhibitor.These results demonstrate some of the limitations of linear scenarios for reaction-diffusion systems and will help to guide projects to engineer synthetic Turing patterns.

ABSTRACTIt is hard to bridge the gap between mathematical formulations and biological implementations of Turing patterns, yet this is necessary for both understanding and engineering these networks with synthetic biology approaches. Here, we model a reaction-diffusion system with two morphogens in a monostable regime, inspired by components that we recently described in a synthetic biology study in mammalian cells.1 The model employs a single promoter to express both the activator and inhibitor genes and produces Turing patterns over large regions of parameter space, using biologically interpretable Hill function reactions. We applied a stability analysis and identified rules for choosing biologically tunable parameter relationships to increase the likelihood of successful patterning. We show how to control Turing pattern sizes and time evolution by manipulating the values for production and degradation relationships. More importantly, our analysis predicts that steep dose-response functions arising from cooperativity are mandatory for Turing patterns. Greater steepness increases parameter space and even reduces the requirement for differential diffusion between activator and inhibitor. These results demonstrate some of the limitations of linear scenarios for reaction-diffusion systems and will help to guide projects to engineer synthetic Turing patterns.

Mentions:
Parameter region for Turing patterns. Region of the parameter spacewhere Turing instabilities develop (i.e., where kmax is positive) for nH =2 and D = 0.01 (A) or D = 0.1 (B).The region of pattern-forming parameter space increases when D decreases (i.e., a 100-fold difference in inhibitor diffusion,relative to slower activator diffusion, is more likely to yield Turingpatterns than is a 10-fold difference). Turing space also increaseswith the steepness of the Hill function nH (see Figures 7 and 8).

Bottom Line:
We applied a stability analysis and identified rules for choosing biologically tunable parameter relationships to increase the likelihood of successful patterning.Greater steepness increases parameter space and even reduces the requirement for differential diffusion between activator and inhibitor.These results demonstrate some of the limitations of linear scenarios for reaction-diffusion systems and will help to guide projects to engineer synthetic Turing patterns.

ABSTRACTIt is hard to bridge the gap between mathematical formulations and biological implementations of Turing patterns, yet this is necessary for both understanding and engineering these networks with synthetic biology approaches. Here, we model a reaction-diffusion system with two morphogens in a monostable regime, inspired by components that we recently described in a synthetic biology study in mammalian cells.1 The model employs a single promoter to express both the activator and inhibitor genes and produces Turing patterns over large regions of parameter space, using biologically interpretable Hill function reactions. We applied a stability analysis and identified rules for choosing biologically tunable parameter relationships to increase the likelihood of successful patterning. We show how to control Turing pattern sizes and time evolution by manipulating the values for production and degradation relationships. More importantly, our analysis predicts that steep dose-response functions arising from cooperativity are mandatory for Turing patterns. Greater steepness increases parameter space and even reduces the requirement for differential diffusion between activator and inhibitor. These results demonstrate some of the limitations of linear scenarios for reaction-diffusion systems and will help to guide projects to engineer synthetic Turing patterns.